## Nonlinear problems of hydrodynamics

### Syllabus

1. Theoretical mechanics of an ideal fluid
1. The least action principle in hydrodynamics. Lagrangian and Eulerian description of fluid motion. The continuity equation. Functionals and variational derivatives. The variational Euler-Lagrange equation for fluid elements. The relabeling symmetry of a fluid medium and actual dependence of the Lagrangian functional on the Eulerian fields only. Examples of Lagrangians: the usual hydrodynamics, the relativistic hydrodynamics, the two- fluid nonrelativistic plasma model, the electron magnetic hydrodynamics. General structure of the equations of motion in the Eulerian description - the generalized Euler equation.
2. Hamiltonian formalism in the fluid mechanics. Definition of the canonical momentum of a fluid element. The Hamiltonian functional. Degenerated non-canonical Poisson bracket. The helicity functional - an example of a Casimir invariant. The frozen-in property of the generalized vorticity. The Kelvin theorem about conservation of circulation and the Cauchy's invariant as consequences of the relabeling symmetry. Variational principles determining the dynamics of frozen-in vortex structures of a given topology. The Clebsch variables as an example of canonical variables. The Lagrangian of the system in the vortex line representation.
3. Potential flows.
1. The sound. The Hamiltonian of sound waves. Quadratic approximation and the normal variables. Nonlinear effects, calculation of the matrix elements of nonlinear wave interaction.
2. One-dimensional gas dynamics. Characteristics. The Riemann invariants. The Hodograph method. Breaking of simple waves. Shock waves. Discontinuities in the initial conditions. The "shallow water" theory. The Burgers equation.
3. Waves at the free surface. General form of the Lagrangian for incompressible potential flows with a free surface. Bubbles and drops. Canonical variables η and ψ . The dispersion relation for surface waves. Asymptotic expansion of the Hamiltonian in powers of the small nonlinearity parameter. Competition between dispersion and nonlinearity in traveling wave, the Korteweg-de-Vries equation, solitons. Conformal variables in the planar problem with a free boundary. The problem of exciting of waves by the wind.
4. Nonlinear resonant wave interaction. Reduction of Hamiltonians. The n-wave problem. Explosive instability. Nonlinear Schroedinger equation for wave envelope of weakly nonlinear quasi-monochromatic wave. Wave collapses. Weakly super-critical instabilities, formation of structures. Wave turbulence. Kinetic equation. Cascades of energy and wave action.
4. Vortex structures in ideal fluid. Vortex sheets, instability of a tangent discontinuity. Two-dimensional flows with piecewise constant vorticity. Thin vortex filaments in 3D space and point vortices in 2D plane. The Hamiltonian and dynamics of point vortices, application of the theory of analytical functions. The Hasimoto transform and integrable local induction approximation in a single vortex filament dynamics. The dispersion law for small perturbations of a straight vortex filament. The Crow instability for two anti-parallel vortex filaments. The problem of finite-time singularity formation from smooth initial data in the Euler equation.
2. Viscous fluid
1. Laminar flows. Equations of motion with taking into account dissipative processes - the Navier-Stokes equations. The energy dissipation. The flow in pipe. The fluid motion between two rotating cylinders. The similarity law. Low-Reynolds-number flows. Flow around sphere, the Stokes formula. Flow around cylinder, Oseen equation. Laminar trail. Laminar boundary layer, Prandtl equations. Thermal conductivity in a fluid. Free convection. Convective instability of fluid.
2. Turbulence. The problem of stability of stationary fluid flow. Chaos in dynamical systems, strange attractor. Developed turbulence. The Kolmogorv's ideas about cascade. Correlation functions of the velocity field. Abnormal dimensionality.
3. Different examples of hydrodynamical phenomena
1. Superfluid hydrodynamics. Spectrum of elementary excitations in a quantum Bose- fluid and the super-fluidity phenomenon. Two-velocity hydrodynamics. Quantized vortex filaments. Vortex grid, Tkachenko waves. Superfluid turbulence. Weakly-non-ideal Bose gas at zero temperature, the Gross-Pitaevskii equation. Instability of the condensate and collapse of the wave function in the attraction case.
2. Plasma dynamics.
1. Kinetic plasma description. Collisionless plasma. Self-consistent field. Vlasov equation. Various kinds of waves in plasma. Landau damping. Relaxation of an initial perturbation. "Echo" in plasma. Adiabatic capture of electrons. Particle collisions in plasma. Instabilities in plasma.
2. Hydrodynamic plasma description. Many-component hydrodynamical plasma models and their different limit cases: quasi-neutral plasma, magnetic hydrodynamics (MHD), Hall MHD, electron MHD.